Question: $P(x)=x^4-2x^3+kx-4$ where $k$ is an unknown integer. $P(x)$ divided by $(x-1)$ has a remainder of $0$. What is the value of $k$ ? $k=$
We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x-{1})$, is equal to $P({1})$. We also know that this remainder is equal to $0$. Therefore, $P({1})=0$. We can use this equality to find $k$. $\begin{aligned} P({1})&=0 \\\\ ({1})^4-2({1})^3+k({1})-4&=0 \\\\ 1-2 \cdot 1+k\cdot 1-4&=0 \\\\ -5+k&=0 \\\\ k&=5 \end{aligned}$ In conclusion, $k=5$.